
theorem Th7: :: EXAMPLES 4.11.(2)
  for X,E be set for L be CLSubFrame of BoolePoset X holds E in the
carrier of CompactSublatt L iff ex F be Element of BoolePoset X st F is finite
  & E = meet { Y where Y is Element of L : F c= Y } & F c= E
proof
  let X,E be set;
  let L be CLSubFrame of BoolePoset X;
A1: the carrier of L c= the carrier of BoolePoset X by YELLOW_0:def 13;
A2: L is complete LATTICE by YELLOW_2:30;
  thus E in the carrier of CompactSublatt L implies ex F be Element of
BoolePoset X st F is finite & E = meet { Y where Y is Element of L : F c= Y } &
  F c= E
  proof
A3: (closure_op L).:([#]CompactSublatt (BoolePoset X)) = [#]CompactSublatt
    Image (closure_op L) by WAYBEL_8:25
      .= [#]CompactSublatt (the RelStr of L) by WAYBEL10:18
      .= the carrier of CompactSublatt (the RelStr of L);
    assume E in the carrier of CompactSublatt L;
    then E in (closure_op L).:([#]CompactSublatt (BoolePoset X)) by A2,A3,Th5;
    then consider x be object such that
A4: x in dom (closure_op L) and
A5: x in [#]CompactSublatt (BoolePoset X) and
A6: E = (closure_op L).x by FUNCT_1:def 6;
    reconsider F = x as Element of BoolePoset X by A4;
    id(BoolePoset X) <= closure_op L by WAYBEL_1:def 14;
    then id(BoolePoset X).F <= (closure_op L).F by YELLOW_2:9;
    then F <= (closure_op L).F;
    then
A7: F c= (closure_op L).F by YELLOW_1:2;
    (closure_op L).x in rng (closure_op L) by A4,FUNCT_1:def 3;
    then (closure_op L).x in the carrier of Image (closure_op L) by
YELLOW_0:def 15;
    then (closure_op L).x in the carrier of the RelStr of L by WAYBEL10:18;
    then
A8: (closure_op L).x in { Y where Y is Element of L : F c= Y } by A7;
    take F;
    F is compact by A5,WAYBEL_8:def 1;
    hence F is finite by WAYBEL_8:28;
A9: (uparrow F) /\ the carrier of L c= { Y where Y is Element of L : F c= Y }
    proof
      let z be object;
      assume
A10:  z in (uparrow F) /\ the carrier of L;
      then reconsider z9 = z as Element of BoolePoset X;
      z in uparrow F by A10,XBOOLE_0:def 4;
      then F <= z9 by WAYBEL_0:18;
      then
A11:  F c= z9 by YELLOW_1:2;
      z in the carrier of L by A10,XBOOLE_0:def 4;
      hence thesis by A11;
    end;
    { Y where Y is Element of L : F c= Y } c= (uparrow F) /\ the carrier of L
    proof
      let z be object;
      assume z in { Y where Y is Element of L : F c= Y };
      then consider z9 be Element of L such that
A12:  z = z9 and
A13:  F c= z9;
      reconsider z1 = z9 as Element of BoolePoset X by A1;
      F <= z1 by A13,YELLOW_1:2;
      then z in uparrow F by A12,WAYBEL_0:18;
      hence thesis by A12,XBOOLE_0:def 4;
    end;
    then
A14: (uparrow F) /\ the carrier of L = { Y where Y is Element of L : F c=
    Y } by A9,XBOOLE_0:def 10;
    thus
A15: E = "/\"((uparrow F) /\ the carrier of L,BoolePoset X) by A6,
WAYBEL10:def 5
      .= meet { Y where Y is Element of L : F c= Y } by A8,A14,YELLOW_1:20;
    now
      let v be object;
      assume
A16:  v in F;
      now
        let V be set;
        assume V in { Y where Y is Element of L : F c= Y };
        then ex V9 be Element of L st V = V9 & F c= V9;
        hence v in V by A16;
      end;
      hence v in E by A8,A15,SETFAM_1:def 1;
    end;
    hence thesis;
  end;
  thus ( ex F be Element of BoolePoset X st F is finite & E = meet { Y where Y
  is Element of L : F c= Y } & F c= E ) implies E in the carrier of
  CompactSublatt L
  proof
    given F be Element of BoolePoset X such that
A17: F is finite and
A18: E = meet { Y where Y is Element of L : F c= Y } and
    F c= E;
    F is compact by A17,WAYBEL_8:28;
    then
A19: F in [#]CompactSublatt (BoolePoset X) by WAYBEL_8:def 1;
A20: (uparrow F) /\ the carrier of L c= { Y where Y is Element of L : F c= Y }
    proof
      let z be object;
      assume
A21:  z in (uparrow F) /\ the carrier of L;
      then reconsider z9 = z as Element of BoolePoset X;
      z in uparrow F by A21,XBOOLE_0:def 4;
      then F <= z9 by WAYBEL_0:18;
      then
A22:  F c= z9 by YELLOW_1:2;
      z in the carrier of L by A21,XBOOLE_0:def 4;
      hence thesis by A22;
    end;
    { Y where Y is Element of L : F c= Y } c= (uparrow F) /\ the carrier of L
    proof
      let z be object;
      assume z in { Y where Y is Element of L : F c= Y };
      then consider z9 be Element of L such that
A23:  z = z9 and
A24:  F c= z9;
      reconsider z1 = z9 as Element of BoolePoset X by A1;
      F <= z1 by A24,YELLOW_1:2;
      then z in uparrow F by A23,WAYBEL_0:18;
      hence thesis by A23,XBOOLE_0:def 4;
    end;
    then
A25: (uparrow F) /\ the carrier of L = { Y where Y is Element of L : F c=
    Y } by A20,XBOOLE_0:def 10;
    id(BoolePoset X) <= closure_op L by WAYBEL_1:def 14;
    then id(BoolePoset X).F <= (closure_op L).F by YELLOW_2:9;
    then F <= (closure_op L).F;
    then
A26: F c= (closure_op L).F by YELLOW_1:2;
    F in the carrier of BoolePoset X;
    then
A27: F in dom (closure_op L) by FUNCT_2:def 1;
    then (closure_op L).F in rng (closure_op L) by FUNCT_1:def 3;
    then (closure_op L).F in the carrier of Image (closure_op L) by
YELLOW_0:def 15;
    then (closure_op L).F in the carrier of the RelStr of L by WAYBEL10:18;
    then (closure_op L).F in { Y where Y is Element of L : F c= Y } by A26;
    then E = "/\"((uparrow F) /\ the carrier of L,BoolePoset X) by A18,A25,
YELLOW_1:20
      .= (closure_op L).F by WAYBEL10:def 5;
    then
A28: E in (closure_op L).:([#]CompactSublatt (BoolePoset X)) by A27,A19,
FUNCT_1:def 6;
    (closure_op L).:([#]CompactSublatt (BoolePoset X)) = [#]
    CompactSublatt Image (closure_op L) by WAYBEL_8:25
      .= [#]CompactSublatt (the RelStr of L) by WAYBEL10:18
      .= the carrier of CompactSublatt (the RelStr of L);
    hence thesis by A2,A28,Th5;
  end;
end;
